The mode spectrum of micrometer-sized ferromagnetic Permalloy disks, exhibiting a vortex ground state, is investigated by means of time-resolved scanning Kerr microscopy. The temporal evolution of the magnetization is probed after application of a fast in-plane field pulse. The lowest order azimuthal mode, a mode with only one diametric node, splits into a doublet as the disk diameter decreases. Theoretical models show that this splitting is a consequence of the interaction of the mode with the gyrotropic motion of the vortex core. Our experiments and micromagnetic simulations confirm that by removing the vortex core from the disk, the mode splitting vanishes.